# A Course on Borel Sets

Springer Science & Business Media, Apr 13, 1998 - Mathematics - 261 pages
A Course on Borel sets provides a thorough introduction to Borel sets and measurable selections and acts as a stepping stone to descriptive set theory by presenting important techniques such as universal sets, prewellordering, scales, etc. It is well suited for graduate students exploring areas of mathematics for their research and for mathematicians requiring Borel sets and measurable selections in their work. It contains significant applications to other branches of mathematics and can serve as a self- contained reference accessible by mathematicians in many different disciplines. It is written in an easily understandable style and employs only naive set theory, general topology, analysis, and algebra. A large number of interesting exercises are given throughout the text.

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### Contents

 Cardinal and Ordinal Numbers 1 12 Order of Inﬁnity 4 13 The Axiom of Choice 7 14 More on Equinumerosity 11 15 Arithmetic of Cardinal Numbers 13 16 WellOrdered Sets 15 17 Transﬁnite Induction 18 18 Ordinal Numbers 21
 44 The First Separation Theorem 147 45 OnetoOne Borel Functions 150 46 The Generalized First Separation Theorem 155 47 Borel Sets with Compact Sections 157 48 Polish Groups 160 49 Reduction Theorems 164 410 Choquet Capacitability Theorem 172 411 The Second Separation Theorem 175

 19 Alephs 24 110 Trees 26 111 Induction on Trees 29 112 The Souslin Operation 31 113 Idempotence of the Souslin Operation 34 Topological Preliminaries 39 22 Polish Spaces 52 23 Compact Metric Spaces 57 24 More Examples 63 25 The Baire Category Theorem 69 26 Transfer Theorems 74 Standard Borel Spaces 80 32 BorelGenerated Topologies 91 33 The Borel Isomorphism Theorem 94 34 Measures 100 35 Category 107 36 Borel Pointclasses 115 Analytic and Coanalytic Sets 127 42 𝚺11 and 𝚷11 Complete Sets 135 43 Regularity Properties 141
 412 CountabletoOne Borel Functions 178 Selection and Uniformization Theorems 183 51 Preliminaries 184 52 Kuratowski and RyllNardzewskis Theorem 189 53 Dubins Savage Selection Theorems 194 54 Partitions into Closed Sets 195 55 Von Neumanns Theorem 198 56 A Selection Theorem for Group Actions 200 57 Borel Sets with Small Sections 204 58 Borel Sets with Large Sections 206 59 Partitions into G𝜎 Sets 212 510 Reflection Phenomenon 216 511 Complementation in Borel Structures 218 512 Borel Sets with σCompact Sections 219 513 Topological Vaught Conjecture 227 514 Uniformizing Coanalytic Sets 236 References 241 Glossary 250 Index 253 Copyright

### Popular passages

Page 244 - Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser., 128, Cambridge Univ.

### About the author (1998)

Srivastava-Indian Statistical Institute, Calcutta, India